N A S A C O N T R A C T O R

R E P O R T

N A S A C R - 2 3 1 9

CO

A STUDY OFCRACK CLOSURE IN FATIGUE

by T. T. Shih and R. P. Wei

Prepared by

LEHIGH UNIVERSITY

Bethlehem, Pa.

for Langley Research Center

NATIONAL A E R O N A U T I C S AND SPACE ADMINISTRATION • WASHINGTON, D. C. • OCTOBER 1973

https://ntrs.nasa.gov/search.jsp?R=19730024067 2018-06-23T09:35:20+00:00Z

1. Report No.

NASA CR-2319

2. Government Accession No.

4. Title and Subtitle

A STUDY OF CRACK CLOSURE IN FATIGUE•f •" ;••'if- *.'..,.

' •' -' !~3.' •,,-•;;;•. , ,,.,.-,

7. Author(s)

T. T. Shih, and R. P. Wei

9. Performing Organization Name and Address

Department of Mechanical EngineLehigh UniversityBethlehem, PA

Bring and Mechanics

12. Sponsoring Agency Name and Address

National Aeronautics and Space AdministrationWashington, D.C. 20546

3. Recipient's Catalog No.

5. Report Date

October 19736. Performing Organization Code

8. Performing Organization Report No.

IFSM-72-25

10. Work Unit No.

501-22-02-01

11. Contract or Grant No.

NGL-39-007-040

13. Type of Report and Period Covered

Contractor Report

14. Sponsoring Agency Code

15. Supplementary Notes

This is a final report.

16. Abstract

Crack closure phenomenon in fatigue was studied by using a Ti-6Al-4V titanium alloy. The

occurrence of crack closure was directly measured by an electrical-potential method, and indirectly

by load-strain measurement. The experimental results showed that the onset of crack closure

depends on both the stress ratio, R, and the maximum stress intensity factor, K . No crack

closure was observed for stress ratio, R, greater than 0.3 in this alloy.

A two-dimensional elastic model was used to explain the behavior of the recorded load-strain

curves. Closure force was estimated by using this model. Yield level stress was found near the

crack tip. Based on this estimated closure force, the crack opening displacement was calculated.

This result showed that onset of crack closure detected by electrical-potential measurement and

crack-opening-displacement measurement is the same.

The implications of crack closure on fatigue crack are considered. The experimental results

show that crack closure cannot fully account for the effect of stress ratio, R, on crack growth,

and that it cannot be regarded as the sole cause for delay.

17. Key Words (Suggested by Author(s))

Fatigue crack growthCrack closureTi-6Al-4V titanium alloy

18. Distribution Statement

Unclassified - unlimited

19. Security Qassif. (of this report)

Unclassified

20. Security Classif. (of this page)

Unclassified

21. No. of

85

22. Price*pomestic, $3-75Foreign, $6.35

For sale by the National Technical Information Service, Springfield, Virginia 22151

TABLE OF CONTENTS

List of Tables ivList of Figures - v

Summary 1

List of Symbols 2

I. INTRODUCTION 4II. MATERIAL AND EXPERIMENTAL WORK 8

A. Material and Specimen 8B. Crack Monitoring System 9C. Environment Control System 11D. Experimental Work 11

1. Electrical Potential Measurements 122. Strain Measurements 13

III. RESULTS AND DISCUSSION 14A. Electrical Potential Measurements 14B. Strain Measurements 16C. The Effects of Stress Ratio and Kmax on Crack Closure 18D. Discussion 19

IV. MODELING FOR CRACK CLOSURE 21A. Analytical Model 21B. Behavior of Load-Strain Curves 24

1. Strain Gage Located Ahead of the Crack Tip 242. Strain Gage Located Just Behind the Crack Tip 253. Strain Gage Located Behind the Crack Tip 26

C. Estimate of Closure Force 27D. Computation of Crack Opening Displacement 28

V. GENERAL DISCUSSION 29VI. CONCLUSIONS 32

TABLES 33FIGURES 36REFERENCES 74APPENDIX 77

iii

LIST OF TABLES

Page

TABLE I Chemical Composition and Tensile Properties 33

TABLE II Strain Gage Position 34

TABLE III Estimated Closure Stress 35

IV

LIST OF FIGURES

Figure Page

1 Schematic Illustration of Load versus Crack-OpeningDisplacement Curve 36

2 Center-Cracked Specimen 3?

3 Potential Leads on a Center-Cracked Specimen 38

4 Schematic Diagram of Electrical-Potential System forCrack Measurement 39

5 Schematic Diagram of Environmental Control System 40

6 Load versus Electrical Potential Curves (Before theCompressive Loading Cycle) 41

7 Load versus Electrical Potential Curves (for the CompressiveLoading Cycle) 45

8 Load versus Electrical Potential Curves (after theCompressive Loading Cycle) 50

9 Load versus Electrical Potential Curve for an IdealizedCrack in an Elastic Medium 51

10 Load versus Strain Curve for an Idealized Crack in anElastic Medium 52

11 Load versus Strain Curves 53

12 Relationship between U and Kmax f°r Different Stress Ratios 58

13 Superposition Model 60

14 Central Crack of Length 2a in an Infinite Plate with EqualPairs of Splitting Forces at x = +b 61

15 Relationships between Strains and the Position of Unit SplittingForces with Strain Gages Located Ahead of the Crack Tip(a = 0.34 in) 62

16 Relationships between Strains and the Position of UnLt SplittingForces with Strain Gage Located Behind the Crack Tip(a = 0.908 in) 63

17 Typical Load versus Strain Curve for a Strain GageLocated Ahead of the Crack Tip 64

18 Typical Load versus Strain Curve for a Strain GageLocated Just Behind the Crack Tip 65

19 Longitudinal Strain Calculatedby the SuperpositionModel and the Estimated Closure Force 66

Page

20 Comparison of Probable Closure Line with Closure LineIndicated by the Electrical Potential Measurements

21 The Difference between the Plastic Zone Size Near theSurface and in the Center

22 Relationship between Load and Relative DisplacementBetween Gage Points Computed by Using the Superposition 69Model

23 Schematic Shown of the Onset of Crack Closure Detected byThree Different Experimental Techniques

24 Schematic Diagram Illustrating Effective AK forSeveral Cases

25 Crack Growth Rate for Ti-6Al-4V Alloy 72

26 The Effect of Kmtn on Delay 73

VI

A STUDY OF CRACK CLOSURE IN FATIGUE

byT. T. Shih and R. P. Wei

LEHIGH UNIVERSITYBethlehem, Pennsylvania

SUMMARY

Crack closure phenomenon in fatigue was studied by using a TI-6A1-4V titaniumalloy. The occurrence of crack closure was directly measured by an electrical-potentialmethod, and indirectly by load-strain measurement. The experimental results showedthat the onset of crack closure depends on both the stress ratio, R, and the maximumstress intensity factor, Kmax. No crack closure was observed for stress ratio, R,greater than 0.3 in this alloy.

A two-dimensional elastic model, was used to explain the behavior of the recordedload-strain curves. Closure force was estimated by using this model. Yield levelstress was found near the crack tip. Based on this estimated closure force, the crackopening displacement was calculated. This result showed that onset of crack closuredetected by electrical-potential measurement and crack-opening-displacement measure-ment is the same.

The implications of crack closure on fatigue crack are considered. The experi-mental results show that crack closure cannot fully account for the effect of stressratio, R, on crack growth, and that it cannot be regarded as the sole cause for delay.

LIST OF SYMBOLS

a half crack lengthaclose distance between the center of the crack and the contact line

aQ initial half crack length

Aa/A,N rate of fatigue crack growth

A empirical constant

b location of crack splitting force with respect to specimen centerline

B specimen thickness

C empirical constant

E Young's Modulus

f (b) the magnitude and distribution of closure force

K stress intensity factor

Kc the critical stress intensity for failure or fracture toughness

K^n^ the maximum stress intensity factor in a loading cycleIJ-lcLX

Kmean the mean value of stress intensity factor in a loading cycleKmin the minimum stress intensity factor in a loading cycle

AK the range of stress intensity factor in a loading cycle; AK = Kmax - Kr

AKeff the effective range of stress intensity factor in a loading cycle

n empirical constant

P applied load

Pmax the maximum applied load in a loading cyclepmin the minimum applied load in a loading cycle

P the applied load above which the crack is fully open

R stress ratio; R = Smin/Smax = Kmin/Kmax

S applied (gross-section) stress; S = P/BW

Smax the maximum applied stress in a loading cycle

Sm- the minimum applied stress in a loading cycle

SQD crack opening stress above which the crack tip is fully open

AS the range of applied stress in a loading cycle; S = Smax - S

AS jf effective stress range

U effective stress or stress intensity range ratio; U =

v y direction displacementvclosure y direction displacement produced by the closure forcevremote y direction displacement produced by remote loadingvsplitting y-direction displacement produced by two pairs of unit splitting force applied

to the crack faces

V electrical potential between the potential leads

V(a) electrical potential between the potential leads

V(aQ) electrical potential between the potential leads corresponding to half-cracklength a0

W gross specimen width

Z(z), Z(z), Z(z), Z'(z) stress function and its derivatives

£ longitudinal strain

(€y)cjogure longitudinal strain produced by closure force

(£y)remote longitudinal strain produced by remote loading

(^•y)splitting longitudinal strain produced by two pairs of splitting force applied to thecrack faces

\) Poisson ratio

cr applied stress

crx longitudinal stress

cry transverse stress

erys yield strength

Txy shear stress

<j)(x,y) Airy's stress function

vf. harmonic functions; i = 1, 2, 3, 4

I. INTRODUCTION

The importance of fatigue in determining the serviceable life of engi-

neering structures has been well recognized [1], In modern high-performance

structure designed for finite service life, fatigue crack growth occurs over a

significant portion of the useful life of the structure. Information on the

kinetics of fatigue crack growth, therefore, becomes an important aspect of

material selection and design.

It has been shown that the crack tip stress intensity factor, K, defined

by linear elasticity, is the most appropriate parameter for characterizing

crack growth [2,3,4], Several empirical relationships for correlating fatigue

crack growth data have been suggested [2, 5-8]. Paris et&i [7] and Paris and

Erdogan [8] suggested that the primary variable of importance for fatigue crack

growth is the range of the crack tip stress intensity factor, AK, and that the

rate of fatigue crack growth, A a/ A N, may be described by a relationship given

by Equation 1.

A and n are empirical constants. The mean value, or the maximum value of K

(Kmean or 1^ ) hi a given load cycle was thought to have only a secondary

effect on crack growth. Forman et^l [6] recognized that the maximum value

of K, Kmax, can have a significant effect on fatigue crack growth, particularly

at maximum K levels approaching KQ (the critical stress intensity for failure).

They suggested that Equation 1 be modified so that the rate of fatigue crack

growth would approach infinity as Kmax approaches K , Equation 2.

>nAa _ A ( A K ) (2)AN l-RK,- AK

The stress ratio, R, is the ratio between the minimum and maximum K levels

in a load cycle; R = Kmm/Kmax. Reasonable correlation with fatigue crack

growth data in the range of 10~6 to 10~3 inch per cycle (2. 54 x 10~5 to 2.54 x

10 mm per cycle) has been demonstrated [9],

The implicit assumptions in both Equations 1 and 2 are (a) that only the

tensile portion of the load cycle is effective in producing fatigue crack growth,

and (b) that there is no crack closure over the tensile portion of the load cycle,

so that the full tensile range of loading is effective. In a recent series of

experiments on a 2024-T3 aluminum alloy, Elber observed that the load versus

crack-opening-displacement curves exhibited a nonlinear region at the lower

load levels, as indicated schematically in Figure 1 [10]. This behavior was

interpreted in terms of crack closure, that is, physical contact between the

fracture surfaces produced by fatigue. Elber suggested that the crack is

closed at the tip over the lower porition of the loading cycles and becomes

open only after the applied stress exceeded a level Sop (Figure 1) and that

fatigue crack growth can occur only during that portion of the loading cycle in

which the crack is fully open. Based on this suggestion, an effective stress

range, ASgg, and an effective stress range ratio, U, were defined.

smax - sop(d;"° °max -

Smax and Sm^n are the maximum and minimum values of the applied stress in

a given cycle; and Sop is the crack opening stress. U can be defined equiva-

lently in terms of the effective stress intensity range, AKeff, and /\K.

Kmax - Kop= ~ - ^7- (4)

^max ^mm

K is the crack opening stress intensity factor corresponding to SQp. Elber

further suggested that the crack growth relationship be written in the following

form:

(AKeff)n = C (UAK)n (5)

Based on a limited range of data, Elber suggested that the effective

stress range ratio, U, is a function of the stress ratio, R, and is independent

°f Smax, or ^max' For the 2024-T3 aluminum alloy sheet, U is given simply

by Equation 6, for R value ranging from -0. 1 to 0. 1.

U = 0.5 + 0.4R -O . l^R^O.7 (6)

Using this empirical result, Elber showed that Equation 5 provided a better fit

to the experimental data than either Equation 1 or Equation 2 [10]. Because

Equation 6 was based on a very limited amount of data and because much of the

data was obtained under conditions of net-section yielding, this relationship for U

should be considered only as a preliminary estimate and must be verified by

additional experimentation. Thus far, no independent verification of this result

has been reported, although some direct evidence for closure has been given by

Buck et al [11-13]. Nevertheless, Elber's concept of effective stress range has

been adopted by several authors [14,15], and the expression for U (Equation 6)

has been accepted and used without question by others [16-18].

The principal purpose of this work is: (a) to provide independent verifi-

cation of the crackclosure phenomenon; (b) to determine the effects of stress

ratio, R, and maximum stress intensity factor, Kmax, on closure; and (c) to

develop an analytical model for estimating the magnitude and distribution of

the closure forces. The experimental and analytical studies are carried out

within the framework of linear-elastic fracture mechanics. A T1-6A1-4V alloy

sheet is used in the experimental studies on closure. Crack closure is mea-

sured directly by means of an electrical potential technique, and indirectly by

measurements of strains in the neighborhood of the crack tip. The implications

of this study to the understanding of several practical problems in fatigue crack

growth are considered.

Acknowledgment is extended to Mr. J. H. FitzGerald for his invaluable

contributions to the experimental work.

II. MATERIAL AND EXPERIMENTAL WORK

A. Material and Specimen

A 0. 2-inch-thick (5. 08 mm) mill annealed Ti-6Al-4V alloy plate was used

in this investigation. The chemical composition, and longitudinal and trans-

verse tensile properties of this alloy are given in Table I.

Three-inch-wide (76.2 mm) by 16-inch-long (406 mm) center-cracked

specimens (Figure 2), oriented in the long transverse (TL) direction, were

used in the fatigue crack growth and crack closure studies. The initial center

notch, about 0.4 inch (about 10 mm) long, was introduced by electro-discharge

machining (EDM). The specimens were precracked in fatigue either at loads to be

used in the subsequent studies, or through a decreasing sequence of loads that

terminated at these loads. The precracking procedure provided fatigue cracks

about 0. 08 inch (about 2mm) in length from the ends of the starter notches cor-

responding to a half-crack length of about 0..3 inch (7. 6 mm), and ensured that

the subsequent tests will be unaffected by the starter notch geometry and by the

residual stress produced by EDM. Both precracking, and the crack growth and.

crack closure experiments were carried out in a 100,000-lb, capacity MTS

system closed-loop electrohydraulic testing machine. Load control was esti-

mated to be better than + 1 percent.

The stress intensity factor, K, for the center-cracked specimen was

computed from Equation 7:

PK = - - /Tta sec (TTa/W) (7 )

where P = applied load, B = specimen thickness, W= gross specimen width,

and a = half crack length. The secant term is a correction for finite specimen

8

width [19] which closely approximates the series correction obtained by Isida

[20].

B. Crack Monitoring System

An electrical potential method was used for monitoring crack growth and

closure utilizing a continuous recording system. This method is based on an

increase in electrical resistance of the specimen with crack growth. A con-

stant current is applied to the specimen, and changes in electrical potential

are measured between fixed points above and below the crack, Figure 3. A

schematic diagram of the system is shown in Figure 4. The relationship

between the electrical potential, V, half crack length, a, and the distance

between the two fixed measurement points, 2y, for the central cracked speci-

men is given by Equation 8:

cosh/cosh TTy/W\

i"1 \cos -fla/W/V (ao) cosh"1 /cosh Try/'

\cos TTa0/W,

This equation, derived by Johnson [21] was used as an analytical calibration

curve. The electrical potential method provides measurements of average

crack length through the thickness, as opposed to the visual methods which

give measurements of the crack length at the specimen surface only. This

method has been shown to agree well with other crack measuring techniques

for a number of materials tested in various environments, provided that crack

tunneling is taken into account [22].

This method of crack measurement has several advantages. It permits

measurements of crack length while the crack is completely covered, thus giving

complete freedom for using environmental chambers which may completely

cover the crack area. Since changes in electrical potential reflect electrical

shorting, vis-a-vis, physical contact, across the crack surfaces, these changes

provide direct measures of crack closure. The other important advantages are

that it gives a continuous measurement of crack length as a function of time and

that it permits continuous recording of load-potential curves. By using a

working current of about 2.1 amperes, this system provided an average mea-

surement sensitivity of about 0. 0025 inch (0. 064 mm) in half crack length, a,

per microvolt (juv) changes in potential (that is, 0. 0025 inch/jiv, or 0. 064 mm

juv) for these specimens. Resolution is better than 0. 001 inch, or 0. 025 mm.

For autographic recording of the load versus potential data used in the

crack closure studies, an alternate amplification procedure was used in place

of the system shown in Figure 4. The electrical potential signal from the

specimen was applied directly to the input of a d-c amplifier in the MTS

Systems testing machine. The output from this amplifier was reduced by a

preset d-c signal, and the difference signal was amplified further in the x-y

recorder and recorded. Because these amplifiers were designed for signal

levels in the millivolt range, and the changes in electrical potential were in

the microvolt range, the background noise tended to be higher with this

method, The reduced signal to noise ratio, however, did not seriously affect

the closure results (see Figure 6, for example) and was acceptable in view of

the improved convenience in measurement.

10

C. Environment Control System

Preliminary experiments indicated that an insulating oxide layer formed

on the fracture surfaces of specimens tested in air. This insulating oxide

layer interferred with the measurement of crack closure by the electrical

potential method, and gave values that underestimated the extent of closure.

To circumvent this problem, all electrical potential measurements of closure

were carried out on specimens tested in dehumidified argon that had been further

purified by a titanium sublimation pump. A schematic diagram of the environ-

ment control system used in these experiments is shown in Figure 5.

Purification was achieved by passing ultrahigh purity argon through a

molecular sieve drier, and a series of cold traps at about -140°C. Further

purification was obtained by passing the argon through a titanium sublimation

pump (TSP) in line with the gas purification system. The TSP was operated

as a getter, and was connected directly to the environment chamber through a

high conductance tube. To reduce possible back diffusion of impurities, the

argon was discharged through an additional cold trap and a silicone fluid trap.

The effectiveness of this purification system has been demonstrated by Wei

and Ritter [23].

D. Experimental Work

Two independent methods were used to study the crack closure phenome-

non. Crack closure was measured directly by using the electrical potential

method. Indirect measurements of crack closure were made by measuring

strains in regions near the crack tip. To simplfy experimentation, only one of

11

the methods was used on each test specimen. The effects of both R and Kmax

were examined.

Closure experiments were carried out, at suitable crack length intervals,

on specimens that have been fatigued under constant load-amplitude cycling at

selected stress ratios, R. Stress ratios of 0. 05, 0.1, 0. 2, 0. 3 and 0.5 were

used with the electrical potential method. For the strain measurements, only

R of 0. 05 was utilized. A constant maximum load, Pmax> of 12,000 Ib

(53. 38 kN) was used for all the tests except for those at R = 0. 05. For the

tests at R = 0. 05, Pmax of 10, 000 Ib. (44.48 kN) was used. Fatigue cracks

were extended from an initial length of about 0. 3 inch to about 0. 9 inch (or,

from about 7.6 to 23 mm). The corresponding values of Kmax ranged from

about 15 to 33 ksi-in^ (16. 5 to 36. 3 MN-m~3/2) for the test at a Pmax of

10, 000 Ib. (44.48 kN), and from about 17 to 40 ksi-in^ (18. 7 to 44 MN-nT3/2)

for tests at Pmov of 12, 000 Ib. (53. 38 kN).IJJLcwC

1. Electrical Potential Measurements

Electrical potential measurements of crack closure were carried

out on specimens that were fatigued in dehumidified argon (see section on

Environment Control System). Fatigue cracks were extended to various selec-

ted lengths under constant load-amplitude cycling at 5 Hz. for prescribed R

and Pmax' Cyclic loading was interrupted at selected crack length intervals,

with the load at Pmax, and the closure experiments were performed. The fol-

lowing sequences were utilized and constitute one set of closure experiments:

(a) the specimen was unloaded from Pmax to Pmui, and then reloaded to Pmax;

12

(b) the specimen was then unloaded from Pmax into compression to about 4, 000

Ib. (17.79 kN compression), and reloaded to Pmax; and (c) finally, step (a) was

repeated. These procedures were carried out using the one-cycle loading feature

on the MTS machine operated at 0. 01 Hz. Autographic recordings of load

versus change in electrical potential (vis-a-vis, change in apparent crack

length - crack closure) were made during each of the aforementioned steps.

2. Strain Measurements

For convenience, specimens for the strain measurements were

fatigued in air. Metal foil strain gages, with 0.015 inch (0.38 mm) gage length,

were used. Up to 4 strain gages were mounted at various locations on the

specimen surface adjacent to the line of intended crack prolongation (see

Table 2 for specific locations for the strain gages). Fatigue cracks were

extended under constant amplitude cyclic loading at 5 Hz., with Pmax =

10,000 Ib. (44.48 kN) and R = 0.05. Cyclic loading was again interrupted at

selected intervals (to approximate those of the electrical potential measure-

ments) for the closure studies. Here, only the load sequence Pmax to 4, 000 Ib.

compression (-17. 79 kN) back to Pmax was used. Autographic recordings of

load versus strain were made during these unloading and reloading cycles.

13

III. RESULTS AND DISCUSSIONS

In this section, the experimental results will be summarized. The

results of the electrical potential and strain measurements will be considered

separately. The effects of stress ratio, R, and maximum stress intensity

factor, Kmax, on crack closure will be discussed. Detailed interpretations of

these experimental results and correlations with an approximate analysis will

be given in the following section on analytical modeling.

A. Electrical Potential Measurements

Typical load versus change in electrical potential curves are shown in

Figure 6-8 for the sets of crack closure experiments described in paragraph

D-l of Section II. Figure 6 depicts those for loading sequence (a) — Pmax to

pmin to pmax; Figure 7, those for sequence (b) — Pmax to -4000 Ib. (-17. 79 kN)

to Pmax; and Figure 8, those for sequence (c) — Pmax to pmin to pmax —

following the sequence into compression. Each of the curves corresponds to

a different crack length, and thus corresponds to a different value of Kmax.

Sensitivity ranged from about 0. 005 in/uv. (0.13 mm/pv.) to 0. 002 in/uv

(0. 05 mm/uv), corresponding to crack lengths, a, of 0. 3 to 0. 8 in. , respectively.

Changes in electrical potential reflects electrical (and, hence physical)

contact between the fracture surfaces, and thus provide a direct measure of

crack closure. For a crack in an elastic medium, the crack surfaces are expected

to be completely separated (open) under an externally applied tensile load, and to

be in complete contact (fully closed) in compression. Load versus change in electrical

14

potential curves for this idealized case are expected to follow the behavior

indicated by Figure 9. In the tensile region, the electrical potential assumes

some value V(a) corresponding to the prevailing crack length. As the applied

load is reduced through zero into compression, the electrical potential under-

goes a step-wise change from V(a), and assumes a value V(ao), corresponding

to that for some initial finite-width notch of length a0, or a value corresponding

to the uncracked specimen. In reality, if crack closure occurs, it is expected

to proceed from the crack tip and extend gradually back towards the initial

notch. This gradual change is reflected in the actual experimental data,

Figures 6-8. Initial deviation from V(a) can be identified with the onset of

crack closure. This point can be identified with the crack opening stress,

Sop, used by Elber [10], and will be utilized in computing U. (Note that, aside

from the identification with the onset of crack closure, no further physical

significance is assumed or implied. Further discussion of this point will be

given in a later section.) The influences of R and Kmax on the onset of crack

closure will be considered separately in a later section.

It is useful to consider the nature of the load versus change in electrical

potential curves in some detail, which serves to qualify the experimental

results and provides some physical insight into the crack closure phenomenon.

It is important to recognize that the electrical potential measurements provide

measures of the average (through the thickness average) crack lengths, and

15

that the identification of a change in electrical potential with an increment of

crack closure is based on the assumption that the electrical conductance across

the fracture faces approximates that of the undamaged material. It is believed

that this assumption was reasonably well met, and was correct in the case of

compression loading based on the agreement between the electrical potential

under compression loading and that of the uncracked specimen. Irrespective

of this assumption, the point for the onset of crack closure can be identified

with good accuracy.

From the experimental data, it can be seen that the paths of the unloading

and reloading curves were somewhat different. The differences are particularly

apparent in those cases where the specimens were loaded into compression.

The differences were caused in part by short-term drift and electrical noise in

the measurement circuit. The principal difference, Figure 7, was produced,

most likely by crushing of the fracture surfaces, and by refracturing of regions

of the fracture surfaces that had become "cold welded" during compression.

B. Strain Measurements

For an idealized crack in an elastic medium, as considered in the pre-

vious section, the load versus strain curves will exhibit two types of behavior

depending on whether the strain gage is located ahead of, or behind the crack

tip. For a gage that is located ahead of the crack tip, the idealized behavior

would be that shown in Figure 10(a). In the tension region, the behavior is

linear and reflects the stress (strain) concentration effect of the crack. In

compression, the slope of the load-strain curve corresponds to that of an

16

uncracked specimen. The abrupt change in slope at zero load reflects the

abrupt change in stiffness as this idealized crack changes from a fully open to

a fully closed configuration. Figure 10(b) illustrates the idealized load-strain

behavior, when the strain gage is located behind the crack tip. In the com-

pression region, the load strain curve again corresponds to that of an uncracked

specimen. In the tension region, since the strain (stress) is essentially unloaded

by the presence of the crack, changes in strain with load become very small.

Similar to the case represented by Figure 10(a), the abrupt change in slope at

zero load is again that associated with opening and closing of the crack.

The load versus strain behavior for cracks in real material is repre-

sented by the curves shown in Figures 11. The load was again cycled from

Pmax into compression, and then back to Pmax- The curves represent mea-

surements made on two separate gages, and reflect changes in load-strain

behavior as the crack tip approached and then moved away from the gage. The

substantial deviations in behavior from those of the idealized crack again reflects

the gradual nature of the closure process. The onset of closure can be detected

most easily when the crack tip is close to the strain gage, and is determined

by the deviation from linear behavior in the tension region. The loads at which

the onset of closure began correlate well with the loads obtained from the

electrical potential measurements. A more detailed discussion of these results

is best made in terms of an analytical model, and will be deferred.

17

C. The Effects of Stress Ratio and Kmax on Crack Closure

On the basis of the electrical potential data, Figures 6 and 7, the effect

of stress ratio, R, and the maximum stress intensity, Kmax, on crack closure

may be considered. Crack closure is considered to begin when the electrical

potential deviates from the value at full load. The point for the onset of crack

closure is consistent with that defined by Elber [10] on the basis of crack opening

displacement measurements. The results are shown in Figure 12, in terms of U

(the effective stress range ratio, U, as defined by Elber). It is to be emphasized

again that U is to be interpreted here only in terms of the onset of crack closure.

No further physical significance is assumed or implied.

U and Pop, corresponding to the onset of crack closure, are shown in

Figures 12a and 12b as functions of Kmax for various R. These results clearly

indicate that the onset of closure is a function of stress ratio, as well as a func-

tion of Kmax and hence of crack length. For R greater than 0.3, no crack closure

was observed (i.e. Pop < Pmm), and U = 1. For values of R below 0.3, U is a

function of Kmax above certain Kmax values, and is independent of Kmax (i.e. U

approaches 1) below these Kmax values, Figure 12a. It is dependent on Kmax in

an intermediate range, and becomes mildly dependent on Kmax at high Kmax

levels. The latter region corresponds to the range in which Pop becomes nearly

constant, Figure 12b. These results are qualitatively consistent with the physical

model (see next section) that suggested an influence of stress ratio, R, and are

consistent with the probable dominance of the surface plastic zones on closure.

18

The results are, however, in contrast with those reported

by Elber [10] for a 2024-T3 aluminum alloy. The differences, of course, can

be attributed in part to the differences in materials. The principal causes,

however, reside in Elber's data. First, because of the limited range of data

(obtained mostly at high Kmax levels) and of the inherent uncertainties in the

results, Elber's assertion that U is independent of Kmax cannot be regarded as

being conclusive. Second, most of Elber's data were obtained

under conditions above net section yielding. As such, the utility of the data and

of the suggested relationship between U and R is questionable. Further independ-

ent verifications are needed.

D. Discussion

Based on the experimental observations, the following processes for

crack closure are envisioned. The physical basis for closure can best be des-

cribed by comparison with idealized cases. For an idealized stationary crack

(infinitesimally thin slit) in an elastic medium, the crack surfaces are expected

to be completely separated (open) under an externally applied tensile load.

The separation between the crack faces (crack opening displacement), can be

defined by linear elastic analysis [24]. Upon unloading to zero load, the crack

opening displacement would return to zero along the full length of the crack.

For an idealized stationary crack in an elastic-plastic medium, the crack

opening displacements (crack contour) under tension are expected to be larger

than those of the elastic case at the same load, as a result of plastic

19

deformation ahead of the crack tip [25], Because of this plastic deformation,

residual crack opening displacements will remain following the reduction to zero

load, and no crack closure would be indicated.

Real cracks, however, are formed as a result of highly localized defor-

mation and separation at the crack tip, and are expected to behave quite dif-

ferently. It is envisioned that the process of deformation and separation (for

example, that suggested by Krafft [26,27]) followed by compaction, would pro-

duce a highly deformed layer. This layer, defined as "a layer of residual ten-

sile strain left in the wake of the crack tip, " [10] delineates an effective crack

contour with crack opening displacements that are, in all probability, smaller

than those of an equivalent elastic crack. On unloading, the surfaces of this

layer may come into contact and give rise to the observed results. Of course,

the degree of compaction depends on the ratio between the minimum and maxi-

mum loads in a cycle, that is, on stress ratio, R. Hence the onset of crack

closure is expected to be a function of R. Since the extent of the residual defor-

mation is dependent on Kmax, the onset of crack closure is expected to be a

function of Kmax, also. The observations of experimental results support this

model.

20

IV. MODELING FOR CRACK CLOSURE

A two-dimensional elastic model is used to assist in the interpretation of

experimental results. It is recognized that such a model cannot truly repre-

sent the real physical problem, which is one that involves elastic-plastic

behavior, with unloading, and is most probably three-dimensional in nature

[28], Nevertheless, it is felt that some semi-quantitative understanding can be

obtained. In the following sections a brief description of the model is given,

and the qualitative features of the model are discussed in terms of the observed

load-strain behavior. On the basis of the model and the experimental data,

estimates of the crack closure forces and the load versus crack opening displace-

ment curves are made and discussed.

A. Analytical Model

The model chosen is that of a notch-like crack in a homogeneous, iso-

tropic elastic body loaded by remote tension and by distributed compressive forces,

representing forces produced by crack closure, on the crack faces as shown in

Figure 13. The magnitude and distribution of the distributed forces are expected to

depend on the degree of closure. It is assumed that the crack faces remain

separated, and the only manifestation of closure is the artifical introduction of

closure forces. It is further assumed that the closure forces are uniformly

distributed through the thicKness (that is, in the z-direction), and vary only in

the x-direction, or along the crack length direction. The problem is then

treated as one of generalized plane stress. Based on these assumptions, the

21

stresses (strains) at any point can be obtained by superposition, Figure 13.

This model is analogous to the strip yield model proposed by Dugdale [29], and

is expected to provide reasonable approximate solutions for stresses (strains)

away from the crack tip. The approximations are expected to be poor, how-

ever, in the near-tip region.

For a plate containing a central through-thickness crack of length 2a, the

strain at any point and the relative displacement between any two points in the

plate may be determined by using the Westergaard method and the appropriate

stress functions [24]. The Westergaard method and the derivation of the rele-

vant strain and displacement equations are summarized in the Appendix. The

longitudinal strain, ^y, and displacement, v, produced by'remote loading on a

finite width plate are given by Equations 9 and 10:

' v'remote" W

a' \ / n a/sec (io)

For the present discussions, these equations are assumed to be valid also for

compression loading, since the crack is assumed to be notch-like and to remain

separated. The longitudinal strain and displacement produced by the distributed

closure forces are obtained by integration of the solutions for two pairs of unit

splitting forces applied to the crack faces, Figure 14. The longitudinal strain

and displacement by the unit splitting forces are given by Equations 11 and 12:

22

££- (ii)w

_ (1+mtm

splitting *

(12)^ _ ;.v. Ha n \\ ft a

The longitudinal strain and displacement produced by the distributed crack

closure forces are then given by Equation 13 and 14:

<Ev>closure=L <Ev>splitting f<b> db (13)/close

/• a(E6,.) , = f (E£ ) ..... f(b) db * *v y closure I a y splitting v

' close

The longitudinal strain and displacement produced by the combination of

remote loading and crack closure forces are then obtained by the superposition

of the solutions given by Equation 15 and 16:

Ev = (Ev)remote+<Ev Closure (15>

E€y= (E £y)remote+ (E £y) closure

The following definitions of terms were used for the previous equations:

z = x + i y ; x,y, coordinate of the point

E, )/ = Young's modulus and Poisson ratio respectively

23

B, W = thickness and width of the specimen, respectively

b = the distance between the center of the crack and splitting force

a close = 'he distance between the center of the crack and the contactline

f(b) = the magnitude and distribution of closure force

Vsec (TTa/W) = correction factor for the finite width specimen

Unfortunately, the magnitude and distribution of the crack closure forces

are not known. Under "the assumptions of the model, an estimate may be made

on the basis of the crack closure data and the load-strain results. Before pro-

ceeding with this estimate, qualitative features of this model may be examined

and discussed in terms of the load-strain results.

B. Behavior of Load-Strain Curves

To facilitate the discussion of the load-strain curves, the longitudinal

strains at fixed gage locations produced by unit splitting forces were computed

as a function of the location of these forces, and are shown in Figures 15 and 16.

Two different crack lengths were chosen. For one, all of the gages were

located ahead of the crack tip; and the other, all behind the crack tip. The

crack tips were sufficiently far away from the gages in both cases such that

the approximations would be reasonable. Some specific cases are discussed

below.

1. Strain Gage Located Ahead of the Crack Tip

A typical load-strain record for a strain gage located ahead of the

crack tip is shown in Figure 17. This curve deviated from the load-strain

24

curve for a notch-like elastic crack whenever the crack closure occurred.

Whenever there is crack closure, compressive forces acting on the crack sur-

faces are expected. These closure forces produce tensile strain at the point

ahead of the crack tip, just like a wedge force acting on the crack (Figure 15).

The tensile strain produced by the closure forces makes the load-strain curves

deviate from the idealized one. Upon further unloading, both the closed area

and the closure force increase, so that the strain-load curve deviates further

away from the idealized curve. The shadowed area depicted in Figure 17 repre-

sents the effect of the closure force.

2. Strain Gage Located Just Behind the Crack Tip

A typical load-strain curve for a strain gage located just behind the

crack tip is shown in Figure 18. Initial deviation from the curve for the notch-

like elastic crack before the onset of crack closure may be attributed to the

onset of reversed yielding [30]. Once closure occurred, the curve deviated

substantially from the idealized one. Now, the closure forces work like dis-

tributed forces acting on a straight boundary, and produce compressive strain

on the gage points. The shadowed area in Figure 18 depicts the influence of the

closure force and the cross hatched area shows approximately the influence of

reversed yielding. Since the strain caused by closure forces and by reversed

yielding are compressive instead of tensile, the shadowed area in Figure 18 is

on the other side of the idealized curve when compared with the shadowed area

in Figure 17.

It should be noted that the rate of decrease of strain was very high just

25

after the onset of crack closure, and slowed down and approached a constant with

further unloading. The initial rapid decrease was caused by both the rapid

increase in closed area and of the closure force effect. As the contact line

moved further away from the strain gage, the closure force on the newly closed

area produced little additional strain on the strain gage. Hence, the rate of

decrease became principally a function of the remotely applied load.

3. Strain Ga%e Located Far Behind the Crack Tip

For constant amplitude loading, the electrical potential measurements

indicated that the onset of crack closure occurred at increasingly higher loads

with increasing Kmax, that is, with increasing crack lengths, Figure I2b.

Strain gage measurements (Figure lib) appear to suggest, however, that the

onset of crack closure occurred at progressively lower loads for the longer

crack lengths. This apparent discrepancy was caused by the reduction in sensi-

tivity when the strain gage is far away from the crack tip and can be explained

in terms of the influence line depicted in Figure 16. Although the onset of crack

closure actually occurred at a higher load for a longer crack length (as indicated

by the electrical potential method), the strain produced at the gage location

(when the crack tip was well past the gage) was small because of its remoteness

from the closure forces near the crack tip. (Of course, the strain produced at

the gage location by reversed yielding at the crack tip would have been small

also). The strain gage could sense the effect of crack closure only when the

contact line approached the gage. This occurred at loads much lower than that

at the onset of closure, as shown in Figure lib. Examination of Figure 11 shows

that the strain produced by the splitting force changed signs depending on the

26

location of the force in relation to the strain gage; being tensile over certain

regions far removed from the strain gage and compressive near the gage. This

change in sign, in combination with changes in magnitude with location, is con-

sistant with the observed behavior.

C. Estimate of Closure Forces

An estimate of the magnitude and distribution of the closure force may be

made on the basis of the superposition model, and experimental data on closure

and on load-strain variations. The estimate was made for the case of R = 0. 05,

since electrical potential measurements of crack closure and companion load-

strain data were available. The extent of crack closure (through-thickness

average) was determined from the electrical potential records, Figure 7(b).

Closure forces with assumed magnitude and assumed distribution over the

closed region were then used, in conjunction with the corresponding remote

loading, to calculate the longitudinal strain (by numerical integration of Equation

16) at the gage location and compared with the experimental load-strain data.

Typically, uniform closure forces (producing yield level stresses) were applied

to a region near the crack tip, and rapidly decaying forces were applied to the

remainder of the closed region. This distribution was considered to be consistent

with the physical model. The forces that produced the best fit with the experi-

mental data (Figure 19) were considered to be a reasonable estimate of the

closure forces. Closure forces determined by this iterative procedure are

shown in Table 3 and appear to be reasonable. A portion of the closure force at

the crack tip may be regarded as a contribution from the compressive yield

zone, although such an identification was not made explicitly.

27

In arriving at these estimates, the distribution of closure forces was

assumed to be constant in the thickness direction, and the crack front and the

closure line were assumed to be straight and perpendicular to the specimen sur-

face. Actually, the contact area would resemble that illustrated in Figure 20.

This assumed three-dimensional distribution is quite reasonable, since the

permanent tensile deformation near the surface is much larger than that in the

center, because of the difference of plastic zone size (Figure 21). In this case,

the distribution of the closure force must be solved as a three-dimensional pro-

blem. Because of this two dimensional approximation, the procedure for esti-

mating the closure forces may be used only when the strain gage is located

close to the crack tip.

D. Computation of Crack Opening Displacement

The relative displacement between several pairs of points in the longitu-

dinal direction (that is, crack opening displacement) were computed by using

the superposition model, and the magnitude and distribution of closure force

estimated previously. The equations used in this computation are Equations 9,

11, 13, and 15.

The computed displacements are depicted in Figure 22. These curves are

similar to those observed by Elber [10], Figure 1, and suggest that the onset of

crack closure detected by several experimental techniques would be the same,

Figure 23.

28

V. GENERAL DISCUSSIONS

It has been shown that the crack tip stress intensity factor, K, defined by

linear elasticity, is the most appropriate parameter for characterizing crack

growth [2, 3,4], Because of a lack of understanding of the mechanism for

fatigue crack growth, no fundamental fatigue crack growth laws have been pro-

posed. The various "so-called" laws are empirical representations of avail-

able experimental data. In proposing the relationship given by Equation 1,

Paris assumed that AK was the most significant parameter, and that stress

ratio, R, (or, Kmean; or 1^ ) is of only secondary importance.* In practi-

cal cases, the effect of R can be important. Attempts have been made to incor-

porate the influence of R into the growth rate relations, again through empiri-

cal correlation [5, 6].

With the observation of crack closure, it is thought that a viable physical

basis for explaining the influence of R on fatigue crack growth had been devel-

oped [10]. Because of closure, the effective stress intensity range, AKe£f,

may differ considerably from the applied AK, and is a function of R. Elber

showed that good agreement with experimental data was obtained on this basis.

Unfortunately, several questions remain unresolved. Firstly, Elber argued

that as soon as crack closure begins, the singularity at the crack tip is removed.

Thus, the corresponding K value at the onset of crack closure is to be regarded

as the KjnJn hi the load cycle; that is, a truncation of the lower portions of the

load cycles. Experimental results from this investigation show the strains ahead of the

* Environmental effects will not be considered in this discussion.

29

crack tip continue to decrease with unloading beyond this point, Therefore, since

fatigue damage is related to the cyclic strain range, the effective Km{n is likely

to be somewhat lower than the level at the onset of closure (Figure 24). These

differences in viewpoint cannot be resolved by the approximate two-dimensional

model developed in this report. Secondly, because of the preliminary nature of

Elber's closure data (see Introduction), the correlation between the rate of

fatigue crack growth and AKeff given by Equation 5 requires additional verifi-

cation . Recent data obtained by FitzGerald and Wei [31] indicate that there is

an effect of R on fatigue crack growth, for R values from 0. 05 to 0. 9, in the

T1-6A1-4V alloy used in this study, Figure 25. The present results indicate

the absence of closure at R greater than 0.3. Hence, crack closure is likely

to be only one of several factors that contribute to the observed R effect.

The importance of delay, or retardation in the rate of fatigue crack growth,

produced by load interactions in variable-amplitude loading on the accurate pre-

diction of fatigue lives has been well recognized for some time [32,33,34],

Jonas and Wei [35] showed that the phenomenon of delay is very complex and

can depend on a broad range of loading variables. Due to a lack of physical

and phenomenological understanding of the effects of load interactions on fatigue

crack growth, no successful model has been proposed to account for delay.

In general, delay effects have been attributed to such things as crack tip

blunting [36], residual stress ahead of the crack tip [34], and more recently,

crack closure proposed by Elber [10]. The first two of these theories do not

lend themselves to direct experimental observation. The third, crack closure

30

proposed by Elber, may be verified experimentally by determining if delay

occurs in the absence of closure. For a loading spectrum depicted on Figure

26, there should not be any crack closure occurring for Km[n equal to or

greater than 12 ksi /in, (R ^ 0.3), according to the result of the last section.

However, significant amount of delay was observed, Figure 26. This

experimental result shows again that crack closure cannot be regarded as the

sole cause for delay.

The present series of experiments have shown that crack closure can and

does occur. Its occurrence is dramatically evidenced by changes in electrical

potential, strain and crack opening displacement with load. However, crack

closure cannot account for all of the observed fatigue behavior. Additional

research is needed to better establish its significance.

31

VI. CONCLUSION

The following conclusions can be made on the basis of the experimental

results on the Ti-6Al-4V alloy:

1. Crackclosure does occur during fatigue. For the TI-6A1-4V alloy,

closure was observed at stress ratios, R, between 0.05 and 0.3.

2. The extent of closure can be determined by electrical-potential mea-

surement method with reasonable accuracy.

3. The stress intensity factor at the onset of closure depends on the stress

ratio, R, and on Kmax.

4. Onset of closure can be detected by measurements of such quantities

as electrical potential, crack-opening displacement, and strain.

5. The closure force near the crack tip was estimated to produce yield

stresses. A part of this estimated closure force may be attributed to the

plastic zone.

6. Crack closure cannot be regarded as the sole cause for the various

observed phenomena for fatigue. For example, it cannot completely account

for the influence of stress ratio, R, on fatigue crack growth; neither can it

fully explain the delay phenomenon under variable-amplitude loading.

32

TABLE 1

Chemical Composition and Tensile Properties(Reactive Metal Ingot No. 293831, Lot 05)

Chemical Composition - Weight Percent(Ingot Analysis)

£ N Fe Al V O H (ppm) Tis

0.03 0.013 0.13 6.2 4.2 0.12 90 Balance

Tensile Properties*

Direction

Transverse

(Average)

Longitudinal

(Average)

Yield Strengthksi (MN/m2)

TensileStrength

ksi (MN/m2)

Elongationin 2 in.

percent

149.5 (1,031)

149.8 (1,033)

148. 3 (1, 023)

149. 2 (1, 029)

141. 1 (973)

141.9 (978)

141.5 (976)

152.6 (1,052)

153.4 (1,058)

151.4 (1,044)

152.5 (1,051)

150.7 (1,039)

151.0 (1,041)

151.8 (1,047)

151.2 (1,042)

12.5

12.5

11.7

12.2

11.7

11.7

12.5

12.0

* Production annealed 1,450°F, 15 min. + air cool.

33

TABLE II

Strain Gage Position(inch)

Position

X

Y

GageA

0.347

0.036

GageB

0.496

0.083

GageC

0.644

0.036

GageD

0.797

0.079

X

Ba DAD ca

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Dia.* I"

Figure 2. Center-Cracked Specimen

37

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Figure 3. Potential Leads on a Center-Cracked Specimen

38

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Figure 6a. Load versus Electrical Potential Curves (beforethe Compressive Loading Cycle) (R = 0.1)

41

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42

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43

SU '0 >- _ ^

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44

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50

LOAD

V ( Q )

V ( Q O )

ELECTRICAL POTENTIAL

Figure 9. Load versus Electrical Potential Curve for anIdealized Crack in an Elastic Medium

51

LOAD

STRAIN

(Q)

LOAD

STRAIN

(b)Figure 10. Load versus Strain Curve for an Idealized Crack

in an Elastic Medium

(a) Strain Gage Located Ahead of the Crack Tip

(b) Strain Gage Located Behind the Crack Tip

52

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Figure 14. Central Crack of Length 2 a in an Infinite Platewith Equal Pairs of Splitting Forces at x = +b

61

x y(a) 0.347 0.036

(b) 0.496 0.083

(c) 0.644 0.036

(d) 0.799 0.079

0.08 0.16 0.24 0.32 0.40b-ln.

Figure 15. Relationships between Strains and the Position of UnitSplitting Forces with Strain Gages Located Ahead ofthe Crack Tip (a = 0.34 in.)

62

x y

(a)0.347 0.036(b) 0.495 0.083(c) 0.644 0.036(d) 0.798 0.079

0.18 0.36 0,54b-in.

0.72 0.90

Figure 16. Relationships between Strains and the Position of UnitSplitting Forces with Strain Gage Located Behind theCrack Tip (a = 0. 908 in.)

63

LOAD (kips)i

3

-2

-2

-3

2 4STRAIN (100 A* in/in)

0 = 0.302x = 0.347y = 0.036

Figure 17. Typical Load versus Strain Curve for a Strain GageLocated Ahead of the Crack Tip

64

0 = 0.349x= 0.347

y = 0.036

-6

CrackClosure

LOAD (kips)

4'

ReverseYielding

-2

-3

STRAIN (100^ in/in)

Figure 18. Typical Load versus Strain Curve for a Strain GageLocated Just Behind the Crack Tip

woa^&

OJO

>> 0)

•° s"5 «I °43 oO T(

C5 W£ WW o

•a s.3 -g

II

oven

66

IndicatedClosure Line

Indicated CrackFront

Actual CrackFront

ProbableClosure Line

( Shaded Area = Aa x b )

Figure 20. Comparison of Probable Closure Line with Closure LineIndicated by the Electrical Potential Measurements

67

0>uo

ooo

<D

N••402

0>

§N

O

rt O

fi w0) QJ

«2 8

I<N

V

I£

68

10

8

W» £+

Q. 6j*

o

Q I Q

0=0.656

x=0.55 y=0.05

x=0.6 y=0.05

x = 0.645 y=0.05

2 4E ( 2 v ) ( l 0 3

8

Figure 22. Relationship between Load and Relative Displacementbetween Gage Points Computed by Using the Super-position Model

69

LJ W

11

1I

L X0

^

11

|1

1|Q.

Q?

\>

111i|i|

UJcr:Dc/)<LU2_j<HZUJ1-On

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J

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o?

LU2UJo:

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71

oo

,LU

(T

Oa:(D

o<a:o

ICT4

10-6

10-7

10'r810 100

AK - ksiyirTFigure 25. Crack Growth Rate for TI-6A1-4V Alloy

72

if)

oCVJ

00co

^^a§.3S

o

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CO<N

CD

ro CJ

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73

REFERENCES

1. H. F. Hardrath, "Fatigue and Fracture Mechanics," AIAA paper No.70-512, April 1970.

2. P. C. Paris, "The Fracture Mechanics Approach to Fatigue1,1 Proc.10th Sagamore Conf., Syracuse University Press (1965).

3. S. R. Swanson, F. Cicci and W. Hoppe, ASTM 416 (1968).

4. H. H. Johnson and P. C. Paris, J. of Eng. Fracture Mech., jL, (1968)

5. D. Broek and J. Schijve, "The Influence of the Mean Stress on the Pro-pagation of Fatigue Cracks in Aluminum Alloy Sheet," NLR-Tech. Rep.M 2111 (1963).

6. R. G. Forman, V. E. Kearney, and R. M. Engle, "Numerical Analysisof Crack Propagation in Cyclically Loaded Structures," J. of Basic Eng.,J59, 3 (Sept. 1967).

7. P. C. Paris, M. P. Gomez and W. E. Anderson, "A Rational AnalyticTheory of Fatigue," The Trend in Engineering, 13, 9, (1961).

8. P. C. Paris and F. Erdogan, J. of Basic Engineering, Trans. ASME,1963.

9. C. M. Hudson and J. T. Scardina, Eng. Fracture Mech., No. 1, (1969).

10. W. Elber, ASTM STP 486 (1971).

11. O. Buck, C. L. Ho, H. L. Marcus and R. B. Thompson, ASTM STP513, (1972).

12. O. Buck, C. L. Ho and H. L. Marcus, reported in "The Symposium onFracture and Fatigue," Washington, D. C. (1972).

13. C. L. Ho, O. Buck and H. L. Marcus, "Application of Strip Model toCrack Tip Resistance andCrack Closure Phenomena," presented atSixth National Symposium on Fracture Mechanics, Philadelphia, Aug.1972.

14. R. A. Schmidt, and P. C. Paris, "Threshold for Fatigue Crack Pro-pagation and the Effects of Load Ratio and Frequency," presented atSixth Annual Fracture Mechanics Symposium, Philadelphia, Aug. 1972.

74

15. M. Katcher and M. Kaplan, "Effect of R Factor and Crack Closure onAluminum and Titanium Alloys," presented at Sixth Annual FractureMechanics Symposium, Philadelphia, Aug. 1972.

16. V. W. Trebules, R. Roberts, and R. W. Hertzberg, "The Effect ofMultiple Overload on Fatigue Crack Propagation in 2024-T3 AluminumAlloy, " presented at Sixth Annual Fracture Mechanics Symposium,Philadelphia, Aug. 1972.

17. R. W. Hertzberg and E. F. J. Von Euw, "Crack Closure and FatigueStriations," Met. Trans. 4, 887 (1973).

18. E. F. J. Von Euw, "Effect of Overload Cycle(s) on Subsequent FatigueCrack Propagation in 2024-T3 Aluminum Alloy, " Ph, D. dissertation,Lehigh University (1971).

19. W. F. Brown, Jr. and J. E. Srawley, ASTM STP 410 (1966).

20. Isida, "Crack Tip Stress Intensity Factor for the Tension of an Eccen-trically Cracked Strip," Lehigh University, Dep. of Mech. Rep. (1965).(see Reference 19)

21. H. H. Johnson, Materials Research and Standards, 5 (1965).

22. C. Y. Li andR. P. Wei, Materials Research and Standards, 6(1966).

23. R. P. Wei and D. L. Ritter, J. of Materials, ASTM (June 1972).

24. P. C. Paris and G. C. Sih, ASTM STP 381 .{l96^).

25. F. A. McClintock and G. R. Erwin, ASTM STP 381 (1965).

26. J. M. Krafft, Trans. ASM,58, 691 (1965).

27. J. M. Krafft, (discussion) ASTM STP 415 (1967).

28. R. P. Wei and T. T. Shih, "Delay hi Fatigue Crack Growth1,1 Int'l J. ofFracture Mechanics (in press).

29. D. S. Dugdale, J. Mech. Phys. Solids, _8 (1960).

30. J. R. Rice, ASTM STP 415 (1967)

31. J. FitzGerald and R. P. Wei, to be published by ASTM (copies are availablefrom the authors).

32. R. H. Christensen, Proceedings - Crack Propagation Symposium,Cranfield, College of Aeronautics (1962).

75

33. J. Schijve, Fatigue Crack Propagation in Light Alloy Sheet Material andStructures, Kept. MP 195, National Luchtvaartlaboratorium (Amsterdam)(August 1960).

34. C. M. Hudson and H. F. Hardrath, NASA TN D-960 (1961).

35. O. Jonas and R. P. Wei, Int'l. J. Fract. Mech., _7 (1971), 116-118.

36. C. M. Hudson and H. F. Hardrath, "Investigation of Variable Loadingon Fatigue Crack Propagation," NASA Tech. Note D-1803 (1965).

76

APPENDIX

The equation of equilibrium for two dimensional elastic problems can be

found in any elementary text of the theory of elasticity.

, 0 A-ldX 9^

Txy - 7y*

The strain-displacement relationships and Hook's law lead to the compatibility

equation.

(-$* i£) (&*<*)• °As is easily checked, equations A-l are satisfied by an Airy's stress function,

<£ (x, y), if the stress components are defined as follows:

A-3

Substitutions of Equation A-3 into Equation A- 2 gives the biharmonic equation

= 0 A 4A'4

For a specific problem, the stress function, <£ , must satisfy Equation A-4 and

the appropriate boundary conditons. Theory of functions of complex variables

provides a powerful tool for the solution of two-dimensional elasticity problems.

77

Define a complex variable, z, by

z = x + i y A-5

If a function of this complex variable, Z(z) > is analytic then its derivatives

exist, that is,g- dz"£j — -.dz

z-§fdZ A-6

,_ dZZ-~dT-

Let Z = ReZ + ilm Z, then the Cauchy Rieman conditions state that:

^— Iml = - ^ ReZ = ImZax

It follows that

V2 (Re!) = V ( I m ) = 0

Similarly, all derivatives of Z have harmonic real and imaginary parts. If

the Yl are each harmonic, that is

y2^- = 0 A-8

then any stress function in the form

A~9

will automatically satisfy the Equation A-4.

In conformity with Equations A-5 - A-9, Westgaard defined an Airy

stress function, 5 , by

78

A"10

This function satisfies the equilibrium Equations, A-l, and the compatibility

condition, Equation A- 2.

Using Equation A-3, A-7, and A-10 the stress are defined by Equation A-ll:

= Re Z - y Im zX

= ReZ + y ImZ' A-ll•7

= - y Re Z 'xy J

The displacement- strain relations are given by Equation A-12:

a /•~ ^* A-12

The strain-stress relations for generalized plane stress condition are given by

Equation A- 13

E£* = <i-^(5A- 13

Combining Equation A-ll, A-12, and A-13, the in-plane displacements and

strains are obtained.

Eu = ( 1 - V) Re Z - ( 1 + V) y Im ZA- 14

Ev = 2ImZ - (1 + V ) y R e Z

79

E6 = (1-V)Re Z - (1+V)y ImZX.

A-15

For a central crack of length 2a in an infinite plate with uniform biaxial stress

at infinity, the stress function is given by Equations A-16 [24]:

Z = 6 z/jz*-tfA-16

The longitudinal strain and displacement at a point A(x, y) are then obtained

from Equations A-14 and A-15.

E6 = ( " !->>) Re•J

A-17

Ev = <S [2 Imjjz2- a2)- (1 +^) y Re

The stress function for a central crack with two equal pairs of splitting forces

at x = +b (Figure 14) is given by Equations A-18.

Z =

Z =

2P , -1-7Ttan ,

2P2 2

a -bA-18

Z = -2P

™» »»—

IT

80

The longitudinal strain and displacement at a point A(x, y) is then given by

Equations A-19.

2P

(z2-b2)/z2-ai

2z2/a2-

(z2- b2)2/z2-a2

a2/a2- b2

(z2-b2)J(z2- a2)3

A-19

Ev = . 2(z2-b2)

NASA-Langley, 1973 32 CR-2319 81

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